2026-10-15 · chapter Yang–Mills and the mass gapPart 14 Closingthe conditional statement, a label for each arrow, the open arrow · Kiichi, Shiori and Rin

The conditional statement — the mass gap and confinement, arrow by arrow

The statement of this chapter is not a single "if … then" but a bundle of arrows. There are two conclusions.

• The mass gap: the spectrum of the transfer matrix (the Hamiltonian) above the vacuum is $\{0\}\cup[\Delta,\infty)$ with $\Delta\gt0$ (4-11).
• Confinement: the expectation value of a large Wilson loop decreases exponentially in the area (the area law, 5-08).

The two are different statements. There is no known road that gets the gap from the area law, and "showing confinement also gives the gap" is a misconception (9-05). So this chapter derives the two conclusions each through its own bundle of arrows, and at the end places, still open, the arrow that joins the two on the same theory.

We also split the levels into two. Level I is the theory on $\mathbb R^2\times$ a small twisted $T^2$ (side $L$, 't Hooft twist $n_{12}=1$), with the limit of lattice spacing $a\to0$ taken. It is not what the problem statement requires but an intermediate target. Level II is the theory on $\mathbb R^4$, reached from Level I through the stage of widening the $T^2$ (the band) and the stage of the infinite-volume limit.

Table 1 lists the assumptions used. Their labels are all Assumption: rows without a proof. The names are taken from the contents; the old symbols are in A-06 §11.

We state the correspondence with the ledger first. In the ledger, [04] and [08] are gathered into a single row ⑥, and the lateral-widening inequality [06] and the uniform slab tension [07] are treated as forms inside the row of the band ⑧. The non-abelian input to the reduction to the band is placed in the ledger in row ⑦, the renormalisation comparison. ⑤, reflection positivity in z, is a closed row in the record, and in this chapter it is used inside arrow C5 (by the rule of the ledger, the row stays until its paper proof is placed in 13-04). [09] and [10] are the differences outside the ledger, Outside 1 and Outside 2; they are not work of removing assumptions but work of carrying the conclusion as far as the problem statement. We put them in the table in this chapter so that they can be named as premises of arrows.

We split the "frame decorrelation" of Table 1 into two forms. [04] is the form for splitting the container estimate on the confinement side, and [05] is the form placed by the research programme on the gap side at Level I. They are assumptions of the same family, but as sentences they are different, and [05] contains the quantity of the conclusion (the rate $m$) (§02). $\theta$ is the bias of the container's sector $\theta(R;T^2_L)$ (side of the container $R$, side of the transverse $T^2$ $L$), a quantity different from the $\theta$ of A-06 Table 6 (the exponent of the accumulation of the shortfall).

Arrow G1 (Level I). Under ultraviolet control [01] and the block form of frame decorrelation [05], the theory on $\mathbb R^2\times$ a small twisted $T^2$ exists (as a subsequential limit of the distributions of block observables), is reflection positive and invariant under translations of $\mathbb R^2$, and its transfer matrix satisfies $\|T_t\|\le e^{-mt}$, $m\ge c/L$ on the orthogonal complement of the vacuum Assumption. This is the form put forward as a proposition by the research programme for Level I, and the proof is at the stage of an outline. The conclusion corresponds to ledger row ②, the gap on a small $T^2$.

To say precisely what this arrow contains: the transfer-matrix step is an abstract lemma of the regulator stage (Stage 1). It is the general fact that, for a self-adjoint contraction on a real inner-product space, if the correlations decay at rate $m$ on a dense set then the operator norm is at most $e^{-m}$; it is independent of the physical measure. So what carries the content of the mass gap inside G1 is the part "decays at rate $m\ge c/L$" of assumption [05]. [05] contains the quantity of the conclusion, and G1 is most accurately read as "an arrow that restates uniform decay of block correlations as a gap of the transfer matrix". That the twist removes the zero modes (the adjoint field has no zero modes, 12-01) is a reason to expect [05] to hold, not a proof of [05].

Arrow G2 (Level II). The gap $m\ge c/L$ of G1 approaches 0 as $L\to\infty$, so as it stands it is not a gap on $\mathbb R^4$. Carrying it to $\mathbb R^4$ needs the mass gap of the slab to be uniform in $L$ [08], and further uses, inside the infinite-volume limit [09], that uniform exponential decay passes to a gap of the transfer matrix of the limiting theory Assumption. Here too, the content of the gap lies in the band assumption [08]. The phrase "the gap comes from Stage 3 and the transfer matrix" describes the case in which the gap of Stage 3 could be carried through the band uniformly in $L$, and the carrying is the problem of the band itself (13-00).

On the confinement side there are two arrows with a paper proof at the stage of fixed lattice spacing; the arrows at the stage of sending the lattice spacing to 0 and at the stage of the band rest on assumptions.

Arrow C1 (fixed lattice spacing). There are two. (a) Slab confinement: with the lattice spacing and the size $N$ of the $T^2$ (in lattice units) fixed, the area law holds for every $\beta$, and the tension is bounded by $\sigma\ge-\ln\tanh(\beta N^2)\approx2e^{-2\beta N^2}$ Proved here (12-04). The tension is set by the cost of the surface of a thin $\mathbb Z_2$ vortex, and approaches 0 in the continuum limit. (b) Theorem A of Paper 3: on the lattice $T^2_N\times T^2_M$ with fixed $N\ge2$, block side $\ell\ge2$, $\beta\ge1$ and $M\in2\ell\mathbb N$, the bias $\theta$ of the $n_{34}$ flux satisfies $-\ln\theta\ge c(N,\ell)\,\beta^{-a}e^{-\beta(S_*(N,\ell)+\varepsilon(\beta))}M^2$, $\varepsilon(\beta)=O(\sqrt{\ln\beta/\beta})$ Assumption (no proof is placed in this book; a proof with citations is in the paper: it cites the chessboard estimate and the Morse–Bott lemma, and the rigidity constants are shown only to exist. The paper is a draft and has not been through outside checking. 12-02 §07 (e)). $S_*$ is the minimal action of a fractional instanton inside a frame frozen to the vacuum, and the statement says that the rate is set by fractional instantons, not by the $2N^2$ of a thin vortex. In the ledger this is the confinement-direction row of ⑦b (the upper end). Theorem B of the same paper is the bound in the opposite direction at fixed $M$. Neither is uniform in $N$, and neither says anything about the continuum limit.

Arrow C2 (Level I). Under ultraviolet control [01] and uniform acceptance of the insertion [02], Wilson loops in the $\mathbb R^2$ plane of the theory on $\mathbb R^2\times$ a small twisted $T^2$ have an area law uniform in $a$ Assumption. The expected form of the tension is $\sigma\ge c'e^{-F/g^2(L)}/L^2$ ($F$ is the prospect of the fractional-instanton action $4\pi^2$). Among the parts of the assembly, the insertion identities and the estimate of the mean are closed on paper and in a finite version in Lean (12-05, 12-06), and in the outline the large-deviation step reduces to a chessboard estimate of the same type as (U2) of [01]. The base point [03] corresponds to the conclusion of this arrow written in the language of the container's bias.

Arrow C3 (the form of the reduction of the band). Under the base point [03], frame decorrelation [04] and the lateral-widening inequality [06], $\theta(R;T^2_R)\le e^{-cR^2}$ Assumption (the conditional theorem of Paper 2). The proof in Paper 2 is a sketch, and the part "frame decorrelation splits the product at the loss of a constant" is not written as a numerical inequality. In this book we rewrite the assumptions in the form of numerical inequalities, and then put the remaining algebra on paper as Theorem 1.

Proved here Proof. Since $2R_k=R_{k+1}$, applying the two parts of (1) in turn gives $t_{k+1}\le K_2\,\theta(2R_k;T^2_{R_k})\le K_1K_2\,t_k^4=Kt_k^4$. Putting $u_k=K^{1/3}t_k$ gives $u_{k+1}\le u_k^4$, and by induction $u_k\le u_0^{4^k}$. Since $4^k=(R_k/R_0)^2$, we have $u_0^{4^k}=\exp(-(R_k/R_0)^2\ln(1/u_0))$, and (2) follows. ∎ Lean Doubling (Doubling.iterate) (assumptions: $c\ge0$, $t_k\ge0$, $t_{k+1}\le c^3t_k^4$; conclusion: $c\,t_k\le(c\,t_0)^{4^k}$), Doubling (Doubling.decay) (the same assumptions and $c\gt0$; conclusion: $t_k\le(c\,t_0)^{4^k}/c$). Read $c=K^{1/3}$.

The starting value $K^{1/3}t_0\lt1$ is made from the base point [03] with a single big jump. If, in the nesting with $T^2_{L_0}$ fixed, $\theta(sL_0;T^2_{L_0})\le C_1(1-\varepsilon_0)^{s^2}$, and, by lateral widening, $\theta(sL_0;T^2_{sL_0})\le C_2s^2\,\theta(sL_0;T^2_{L_0})$ can be said, then the $t_0\le C_1C_2s^2e^{-\varepsilon_0s^2}$ at $R_0=sL_0$ becomes smaller than $K^{-1/3}$ for $s^2\gg\varepsilon_0^{-1}\ln(1/\varepsilon_0)$ Proved here (the same algebra).

Theorem 1 is algebra, and all the content is in (1) and in the two inequalities of the jump. The nesting inequality is exact as a partition inside a container with $T^2$ fixed (12-03), but the sub-containers share frames and are correlated, so splitting into a product with constant $K_1$ needs frame decorrelation [04]. The lateral-widening inequality is [06] itself. That is why the label of C3 is Assumption, and only the part that is Theorem 1 could be put on paper.

Arrow C4 (from the form of the reduction to the area law). From $\theta(R;T^2_R)\le e^{-cR^2}$ and frame decorrelation [04], by the container estimate (the container identity of Mack–Petkova, 12-03 Cited), an area law with tension uniform in $L$ Assumption. The global version that does not use frames uses the ratio of twisted partition functions and the Tomboulis–Yaffe inequality Cited, and in exchange the base point has to be shown globally (13-01).

Arrow C5 (the chessboard form). With the size of the $T^2$ fixed, from reflection positivity in $z$ (ledger ⑤) Proved here (a proposition of Paper 2; ReflPos (ReflPos.refl_pos, ReflPos.refl_cs) is the algebra of the core, 13-04) and the chessboard estimate Cited, frame decorrelation is replaced by a single density $\kappa_{\rm cb}$, and $|\langle W(C)\rangle|\le\kappa_{\rm cb}^{\,n(C)}$ Proved here (a statement with $L$ fixed, 13-05). In this language the band becomes two uniformities, and under the uniform slab tension [07] and the uniform slab gap [08] an area law with tension uniform in $L$ follows Assumption.

Confinement at Level II follows from the area law of C4 or C5 and the limit to $\mathbb R^4$ [09] Assumption.

Arrow J (open). The gap road of §02 and the confinement road of §03 both reach $\mathbb R^4$ through assumptions that cross the band ([08], or [06] and [07]). To state both conclusions on the same theory on $\mathbb R^4$, we need an arrow saying that the two roads give the same limiting theory and that both hold in that theory. This is the joining of the two roads [10] (Outside 2, outside the ledger). Its label is Assumption, but unlike the other assumption rows, there is not yet a sentence saying what it suffices to assume.

Since there is no known road going directly from the area law to the mass gap, J cannot be written as "an extension of the confinement road". At the level of physical expectation the two are linked (excitations on a string of tension $\sigma\gt0$ are heavy, 7-03), but that is not an arrow Reading (a physical estimate). The gap is derived by the road of §02 (carrying the gap of Stage 3 through the band), confinement by the road of §03, and J is limited to the role of saying "the two are statements about the same theory".

The statement of the road from the direction of time. The conditional statement of the road of Part 10 has a different form. From the upper side $\xi^2{\uparrow}$ ($C_P\le K\xi_{\rm RG}^2$) of the diffusive-relaxation-time assumption (formerly Assumption H) and the one-sided bridge (Proposition B, $C_P\ge\kappa\xi^2$), a lower bound on the gap in the physical units of the lattice, $m_{\rm phys}\ge(\kappa/K)^{1/2}$, follows at each $\beta$ Proved here (algebra on top of the assumptions, 10-01 §06). Saying this uniformly as $\beta\to\infty$ is the four-dimensional goal of Part 10 (ledger ⑩) ((T2) of 10-15 Table 5), and this goal too is a statement that crosses the band. The [08] of the road from the direction of space and the "$\xi^2{\uparrow}$ uniform in $\beta$, and the bridge" of the road from the direction of time are uniformity statements of the same type, and neither has been shown inside the band (13-07).

None of the arrows of this chapter shows the mass gap or confinement. The only things whose paper proof could be placed in the book are Theorem 1 (algebra), C1a at fixed lattice spacing, and the first half of C5 with $T^2$ fixed. C1 says nothing about the continuum limit, and C1b (Theorem A of Paper 3) has no proof in the book (the label is Assumption; a proof with citations is in the paper) and is not uniform in $N$. G1 and C2 at Level I are propositions of the research programme, at the stage of an outline. The band forms [06] and [07] contain the quantities of the conclusion (bias, tension) in the form "uniform in $L$", and [08] states decay of correlations in a form uniform in $L$. The statement of the band itself carries the content of the conclusion. The statement of this chapter is a conditional statement only insofar as the band is placed among the assumptions.

Even if every row of the ledger disappears, a difference from what the problem statement requires remains: all compact simple groups (for groups without a centre the vortex tools cannot be used), the OS axioms on $\mathbb R^4$ with the recovery of rotations and non-triviality, and the arrow from the area law to the mass gap. These are Outside 1–4 in the separate frame "the differences that remain outside the ledger" of 14-02 §05 ([09] and [10] of Table 1 are Outside 1 and Outside 2). Nor is the ledger a table that only shrinks: as research advances, rows can split and new rows can appear.

We list the arrows in the format of the five grades (10-00 §01). The label is the label of the arrow, not of its premises. Even for an arrow whose premise is Assumption, the label is Proved here if the inference from the premises to the conclusion is written on paper, and Assumption if it is at the stage of an outline.

Dependency graph (the numbers of the premises are smaller than the number of the conclusion; [01]–[10] are the assumptions of Table 1):

[01] ASM ①       ultraviolet control (U1)(U2)(U3)                     Stage 2
[02] ASM ③       uniform acceptance of the insertion                  Stage 4
[03] ASM ④       base point on a small T²: θ(L₀;T²_L₀) ≤ 1 − ε₀       Stage 4
[04] ASM ⑥fixed  frame decorrelation (container form)                 Stages 4, 5
[05] ASM ②form   frame decorrelation (block form, with rate m ≥ c/L)  Stage 3
[06] ASM ⑧form   lateral-widening inequality (strong / weak form)     Stage 5
[07] ASM ⑧weak   uniform slab tension                                 Stage 5
[08] ASM ⑥       uniform slab gap (frame decorrelation)               Stage 5
[09] ASM Out1    limit to R⁴                                          Stage 6
[10] ASM Out2    joining the two roads (statement not settled)
[11] PRV ⑤       reflection positivity in z                           Lean ReflPos.refl_pos, refl_cs
[12] CIT MP      container identity (Mack–Petkova), Tomboulis–Yaffe inequality
[13] PRV SLAB    C1a slab confinement (lattice spacing and N fixed)   tension → 0 in the continuum limit
[14] ASM THA     C1b Theorem A of Paper 3 (fixed N, β ≥ 1)            proof with citations in the paper
[15] PRV DBL     Theorem 1, the doubling iteration                    Lean Doubling.iterate, decay
[16] PRV CB      C5a T² fixed: |⟨W(C)⟩| ≤ κ_cb^n(C) ⟸ [11]
[17] ASM G1      Level I gap m ≥ c/L ⟸ [01][05]
[18] ASM C2      Level I area law (uniform in a) ⟸ [01][02]
[19] ASM C3      θ(R;T²_R) ≤ e^(−cR²) ⟸ [03][04][06][15]
[20] ASM C4      area law, tension uniform in L (reduction form) ⟸ [04][12][19]
[21] ASM C5b     area law, tension uniform in L (chessboard form) ⟸ [07][08][16]
[22] ASM G2      gap on R⁴ ⟸ [08][09][17]
[23] ASM C6      area law on R⁴ ⟸ [09][20] (or [21])
[24] ASM J       mass gap and confinement on the same R⁴ theory ⟸ [10][22][23]

The cycle check only looks at whether, for every "⟸", the numbers on the right are smaller than the number on the left. Among the assumption rows, those that contain the quantity of the conclusion are [05] (the rate $m$, in a form uniform in $a$) and [06] and [07], the forms of the band ⑧ (bias, tension, in a form uniform in $L$). This is not circular, but it shows that what arrows G1, C3 and C5b add is only a restatement of uniformity and algebra, and that the content lies on the side of these assumptions. The ledger (14-02 §04) writes its rows in forms that do not contain the quantity of the conclusion. [05], [06] and [07], which do contain it, are "restatements" of ledger rows rewritten to fit the arrows of this chapter, not the ledger rows themselves. The content of G2 also lies on the side of [08], the uniform decay of correlations. [13] and [14] are nodes that are not among the ancestors of the conclusion [24]; they record what can be said on paper at the stage of fixed lattice spacing. [18] writes the same content as the base point [03] in the language of Level I, and in the graph we kept [03] as a leaf.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the conditional statement, arrow, Level I and Level II, ultraviolet control, uniform acceptance of the insertion, the base point on a small $T^2$, frame decorrelation, the lateral-widening inequality, the uniform slab tension, the uniform slab gap, joining the two roads, the doubling iteration, the chessboard density $\kappa_{\rm cb}$, the open arrow. Their meanings are in the A-02 Glossary and A-06 §11.

Next chapter to read: 14-02 The ledger of conditions — the present state of each assumption, and what is needed for it to disappear.

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